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Chicago Public Schools, Advanced Algebra Lesson plans, Days 37-53

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  • 134

    STRUCTURED CURRICULUM LESSON PLAN

    Day: 037 Subject: Advanced Algebra Grade Level: 11

    Correlations (SG,CAS,CFS): 6A1

    TAP:Analyze and interpret data presented in charts,

    graphs, tables and other displays

    ISAT:Identify, analyze, and solve problems using

    equations, inequalities, functions, and theirgraphs

    Unit Focus/Foci

    Matrices and Systems of Equations

    Instructional Focus/Foci

    Using Matrices to Store and Represent Data and Adding or Subtracting Matrices

    Materials

    Classroom set of graphing calculatorsOverhead graphing calculatorOverhead projector

    Educational Strategies/Instructional Procedures

    Give the following definitions:

    Matrix: A rectangular arrangement of elements in rows and columns enclosed by brackets.

    Example: M = 1 4 3 2

    3 2 0 5

    LNM

    OQP

    Matrix notation: An entry is named as mab, where a = row number and b = column number

    In the example above, m13= 3; m24=5Dimension: the number of rows x the number of columns gives the dimension of amatrix. The dimension of the matrix M above is 2 x 4.

    Have students place elements in their proper position, then have them enter the followingmatrices into their calculators:

  • 135

    To enter a matrix into the graphing calculator:1. press MATRX2. use the right arrow to EDIT, then ENTER3. enter number of rows, ENTER4. enter number of columns, ENTER5. enter each element, going across. After each element, press ENTER. If you

    make a mistake, scroll to the location you want to change.

    To perform operations on matrices, press MATRX, leave NAMES highlighted and scroll downto the matrix you want to use; press ENTER.

    Students will find: A + B; B A; A A (call this C); C + B; 1 x B

    Students will report findings to the whole class.

    Integration with Core Subject(s)

    LA: Understand explicit, factual informationUnderstand the meaning of words in context

    Connection(s)

    Enrichment: In a recent survey, 90% of high school students were opposed to a systemwideSlice Girls concert suggested by elementary school students.1. Work with a partner to conduct a survey in your school regarding the questions below.

    A. Should the Slice Girls concert occur?B. Should high school students be mandated to attend?C. Should elementary school students be mandated to attend?D. What percent have no opinion?

    Use the following matrix to report your data in percentage form.

    Question Yes No Indifferent

    A B=

    L

    NMMM

    O

    QPPP

    =

    L

    NMMM

    O

    QPPP

    3 2 1 0

    6 2 4 1

    0 3 2 1

    7 3 2 1

    4 0 3 2

    1 7 3 0

    ABCD

  • 136

    Fine Arts:

    Home: Parents will sign homework record weekly.

    Remediation:

    Technology:

    Assessment

    Have students answer the following questions:

    a. What must be true about the dimensions of two matrices that are equal? b. Given matrices A, B, and A + B, why must they have the same dimension? c. What is the relationship between matrices A and 1A? d. What is a good name for matrix C?

    Homework

    Assign appropriate problems from your text.

    Teacher Notes

    Solutions to Assessment:

    a. They must have the same number of rows and columns.b. The entries from the first matrix should correspond to the entries from the second matrix.c. Matrix -1 x A is the opposite of matrix A.d. Zero matrix

    Solutions to in-class assignment:

    A + B =

    11101

    17210

    11110

    B A =

    1541

    3122

    1354

    A A =

    0000

    0000

    0000

    C + B =

    0371

    2304

    1237

    -1 x B

    0371

    2304

    1237

  • 137

    STRUCTURED CURRICULUM LESSON PLAN

    Day: 038 - 039 Subject: Advanced Algebra Grade Level: High School

    Correlations (SG,CAS,CFS): 6A1

    TAP:Perform arithmetic operations involving

    integers, fractions, decimals and percentsexplicitly stated or within context

    Choose and apply appropriate operationalprocedures and problem-solving strategies toreal world situations

    Understand number systemsUse variables, number sentences, and equations

    to represent solutions and solve problems

    ISAT:Solve problems requiring computations with

    whole numbers, fractions, decimals, ratios,percents, and proportions

    Use mathematical skills to estimate,approximate, and predict outcomes and tojudge reasonableness of results

    Identify, analyze, and solve problems usingequations, inequalities, functions, and theirgraphs

    Unit Focus/Foci

    Exploring Matrices and Systems of Equations

    Instructional Focus/Foci

    Performing Matrix Multiplication

    Materials

    Classroom set of graphing calculatorsOverhead graphing calculatorsClassroom set of graphing calculatorsOverhead projector

    Educational Strategies/Instructional Procedures

    Explain to students: To multiply two matrices Amxn and Bpxq, n must equal p (the inner

    dimensions). The resulting matrix AB will have dimension m x q (the outer dimensions). If np, the two matrices cannot be multiplied.

    Example 1: A B=

    LNM

    OQP =

    L

    N

    MMMM

    O

    Q

    PPPP3 2 0 1

    4 3 3 2

    0

    3

    2

    1

  • 138

    dim[A] = 2x4, dim[B] = 4x1, so these two matrices can be multiplied (since 4=4), and theresulting matrix will have dimension 2x1.

    Take row 1 of A and column 1 of B, multiply successive elements, add these products, and theresult is the corresponding element in the product matrix. Using the example above:

    30 + 23 + 02 + 11 = 5; 40 +03 + 32 + -21 = 4. So the product matrix is 5

    4

    LNMOQP

    Give students several matrices of different dimensions and ask if they can be multiplied.

    Example 2:

    a.

    40

    36

    25

    68b.

    82

    43

    96

    41

    72c.

    036

    894

    282

    413

    Solutions: a. yes b. yes c. no

    Have students form small groups; use the graphing calculator to determine whether theassociative, commutative, or distributive properties hold for matrix multiplication.

    Report findings to class. Use the matrices: A =

    06

    14 B =

    63

    12 C =

    20

    31

    a. A*B = B * Ab . A(B*C) = (A * B ) * Cc . A(B+C) = AB + AC

    Example 3:

    Jane and Andrew enjoy shopping for music on a monthly basis. They each purchase thefollowing items.

    CDs Tapes LPsJane 10 6 1

    Andrew 8 2 3

    If CDs cost $15, tapes cost $8, and LPs cost $10, use matrices to find the total cost that Jane andAndrew each spent on music.

  • 139

    Solution:

    328

    1610

    10

    8

    15

    =

    166.

    208

    Integration with Core Subject(s)

    LA: Understand explicit, factual informationUnderstand the meaning of words in context

    Connection(s)

    Enrichment: Use graph paper to draw squares with areas of 2, 5, and 10 units.

    Fine Arts:

    Home: Parents will sign homework record weekly.

    Remediation:

    Technology: Have students review how their calculator can perform various operations withmatrices.

    Assessment

    Teacher observation

    Homework

    Assign appropriate problems from your text.

    Teacher Notes

    This is a two-day lesson. The second day (Day 39) should be used to review homework, forremediation exercises, or for extra practice, as necessary.

    Solutions to in-class assignment:

    a. A*B = B *A (not true) b. A(B*C) = (A *B) * C (true)

    612

    25

    =

    348

    214

    2412

    195=

    2412

    195

    c. A(B+C) = AB + AC (not true)

    invalid dimensions =

    246

    121

  • 140

    STRUCTURED CURRICULUM LESSON PLAN

    Day: 040 Subject: Advanced Algebra Grade Level: High School

    Correlations (SG,CAS,CFS): 8D2

    TAP:Perform arithmetic operations involving

    integers, fractions, decimals and percentsexplicitly stated or within context

    Choose and apply appropriate operationalprocedures and problem-solving strategies toreal world situations

    Understand number systemsUse variables, number sentences, and equations

    to represent solutions and solve problems

    ISAT:Solve problems requiring computations with

    whole numbers, fractions, decimals, ratios,percents, and proportions

    Use mathematical skills to estimate,approximate, and predict outcomes and tojudge reasonableness of results

    Identify, analyze, and solve problems usingequations, inequalities, functions, and theirgraphs

    Unit Focus/Foci

    Exploring Matrices and Systems of Equations

    Instructional Focus/Foci

    Identifying Systems of Two Linear Equations

    Materials

    Handout 4.3Classroom set of graphing calculatorsOverhead graphing calculatorOverhead projector

    Educational Strategies/Instructional Procedures

    Students will: Form groups of 2. Complete Handout 4.3 (exploration on intersections of lines). Report findings to the class.

    Present the following definitions:

    Consistent system: lines intersect in one or more pointsIndependent system: lines intersect in one unique point (one solution)Dependent system: lines intersect in an infinite number of points (the lines coincide; infinitelymany solutions)Inconsistent system: lines do not intersect (parallel lines; no solutions)

  • 141

    Give examples of systems of equations and graphs of solutions of equations.

    Have students match each system with its graph.

    a. y = x + 1 b. y = -2x + 3 c. y = .5x - 2 y = x y = 3x + 1 y = -x + 4

    1. 2.

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